Sir William Rowan Hamilton (1805-1865) His great mathematical intelligence was awakened by the reading of Newton's Arithemtica universalis when he was 12 years old. He went on to the Principia, and when he was 17 attacked Laplace's Mécanique céleste. There he detected an error which was communicated to John Brinkley, then royal astronomer in Ireland, who at once recognised one of the first mathematical minds of the age. Hamilton had a brilliant career at the Trinity college, Dublin, but before it was completed he was appointed, at the age of 22, to the Andrews chair of astronomy in succession to Brinkley. The rest of his life was spent at the observatory at Dunsik, near Dublin, in the close study of mathematics. He married Hellen Mary Bayly in 1833. He was knighted in 1835. At the time of his death, Sept. 2, 1865, Hamilton was working on the Elements of Quaternions, on which the last six years of his life had been spent.
    His earliest papers were the "Theory of Systems of Rays" (Trans. Roy. Irish Acad., 1828-32); in the last of these, by the aid of mathematics, he predicted conical refraction. These were followed by papers on "The Principle of Varying Action" and in 1834 and 1835 by two memoirs "On a General Method in Dynamics". In these, Hamilton defines the action as the path-integral of the Lagrangian function between the initial and final states, so that the physical trajectory is, among all the possible, that one which minimises the action integral, that is, the path of least action. Then, he defines the generalised moments as the partial derivatives of the Lagrangian function L with regard to the velocities and obtains, through a Legendre transform of the Lagrangian function, the H function, which depends on the generalised coordinates and moments instead of the coordinates and velocities. The H function (now called the Hamilton function) is usually the energy of the dynamical system. The two physical revolutions of our finishing century, the Relativity and Quantum theory, would not have been possible without Hamilton's dynamics.
    On Oct. 16, 1843, Hamilton discovered the quaternions while he was walking with his wife along the Royal Canal, cutting with a knife the fundamental formula on a stone of the Brougham Bridge:

        i² = j² = k² = i j k =  -1

    Hamilton's Lectures on Quaternions were published in 1853 and his great book, The Elements of Quaternions, posthumously in 1866. In the preface to the Lectures he describes the steps by which he reached his important conclusions which form the germ of the later basic theories of hypercomplex numbers.
    He left an enormous collection of manuscript books, full of original investigations,  that were handed over to the Trinity college.

Hermann Günther Grassmann (1809-1877) Born in Stettin (at that time a Prussian town, today in Poland), he learned mathematics mainly from his father Justus Günther Grassmann (1779-1852), teacher at the Stettin Gymnasium. He taught science and mathematics at various schools at Stettin despite his attempts to attain a university post. Grassmann is the key man in the development of the geometric algebra. Initially thought and proposed with the name of characteristica geometrica by Leibniz, he could not see the realisation of his idea of a direct calculus of purely geometric entities. In 1844 a prize (45 gold ducats for 1846) was offered by the Fürstlich Jablonowski'schen Gessellschaft in Leipzig to whom was capable to develop the idea of Leibniz. Grassmann won this prize with the Geometric Analysis, published by this society in 1847 with a foreword of August Ferdinand Möbius. Its content is essentially that of Die Ausdehnungslehre (1844), where Grassmann defined the exterior (or outer) product of two or more segments by means of successive parallel transports. His formalism is the groundwork of the geometric algebra. Grassman also worked in statics, dynamics and electromagnetism. He published the Theorie der Ebbe und Flut, the Lehrbuch der Arithmetik, influencing Peano in his system of axioms, and Lehrbuch der Trigonometrie. The scarce reception of the second edition of Die Ausdehnungslehre (1862) caused him to dedicate himself mainly to philological studies. He composed a dictionary of Sanskrit language and translated the Rig-Veda. The article "Der Ort der Hamilton'schen Quaternionen in der Ausdehnungslehre" plays main role in the history of the geometric algebra.

Josiah Willard Gibbs (1839-1903) Born in New Haven (U.S.A.) he graduated in 1858 and attained in 1863 the first doctorate in engineering given in the United States. Then, he became tutor and taught Latin firstly and natural philosophy afterwards at the Yale University until 1866. That year, he went with his sisters to Europe where he spent three years. He studied in Berlin with Magnus, Weierstrass and others and in Heidelberg with Kirchoff and Helmholtz, but he did not visit the United Kingdom. With a good knowledge of the German language, he was imbued by the German way of thinking, which is reflected in all his writings. When he returned to New Haven in 1869, he lived with the family of his sister Julia in the home built by his father until his death. In 1871 Gibbs attained the recently created chair of mathematical physics at Yale college.
    Gibbs was a delayed scientific writer, who applied very widely the mathematical tools to the description of the physical and chemical phenomena. In 1876-78 he published On the Equilibrium of Heterogeneous Substances, where he states the famous rule of phases giving the number of independent thermodynamic magnitudes in a heterogeneous chemical system. From  the differential equation of energy, Gibbs defined mathematically the chemical potential, a new thermodynamic magnitude which could not be directly measured until the apparition of the modern electrodes. I think that the history of thermodynamics has two periods, a first one before Gibbs when the energy, entropy, and other magnitudes were discovered, and a second one after and due to him when thermodynamics became an exact science. From 1877 Gibbs taught electromagnetism and hence he printed privately Elements of Vector Analysis in 1881 and 1884 for the use of his students. He also sent this "pamphlet" to important scientists, so that the quaternionists, whose leader was Peter Guthrie Tait, broke out an epistolary war in Nature. The publication of the book Vector Analysis would wait until 1901. His last work, Elementary Principles in Statistical Mechanics (1903) is  a deep development of  Boltzmann's statistical conception of entropy. On the other hand, Gibbs was the first scientist who rigorously deduced the capillary adsorption equation from thermodynamics.
    Although he was not a mathematician, his mind was of mathematical nature and exact meaning. His mathematical contributions are the Gibbs phenomenon of the Fourier series, a method of determination of an orbit of a comet from only three observations and mainly the vector analysis, which we teach in the same manner and notation that he used.

William Kingdon Clifford (1845-1879). Born in Exeter, he was educated at the King's college, London, and at the Trinity college, Cambridge, where he was elected fellow in 1868. He became professor of mathematics at the University college, London, in 1871. In 1875 he married Lucy Lane, who became well known as a novelist under her married surname. Clifford died prematurely of pulmonary tuberculosis in Madeira. Clifford was simultaneously a philosopher and a mathematician. His philosophic writings are as important as his mathematical papers. He studied non-Euclidean geometry and the biquaternions. Due to his short life, he wrote few but fundamental papers about the geometric algebra. In "Applications of Grassmann's Extensive Algebra" (Am. J. of Math. 1 [1878]) he showed the exact meaning and location of quaternions inside Grassmann's Extension Theory, making the synthesis of both systems which is known today as Clifford algebra. However, Gibbs and Heaviside did not take notice of it. In 1876 Clifford fell ill and he never recovered his health.